# LP Implementation Details

**Reviewer Issue 3.5**: Document exact dependent variable construction.

## Estimation Framework

All local projections follow Jordà (2005):

For **growth variables** (cumulated):
$$y^{cum}_{j,t+h} = \sum_{s=0}^{h} \Delta y_{j,t+s} = \alpha_h + \beta_h x_{j,t} + \gamma_h X_{j,t} + \varepsilon_{j,t+h}$$

For **level variables** (first-differenced from baseline):
$$y_{j,t+h} - y_{j,t-1} = \alpha_h + \beta_h x_{j,t} + \gamma_h X_{j,t} + \varepsilon_{j,t+h}$$

Treatment: $x_{j,t}$ = log(predicted demographic inflows)$_{j,t}$

Controls ($X_{j,t}$): fiscal_bal_gdp, nfa_gdp_lag, log_rel_opw, kaopen

Estimation: PanelGLS with Cochrane-Orcutt AR(1) correction.

Sample: OECD (38 countries, 1990-2024).

## Dependent Variable Construction

| Variable | Label | Transformation | Formula | Notes |
|----------|-------|----------------|---------|-------|
| gross_fixed_investment_gdp | Investment/GDP | Level lead | y_{j,t+h} − y_{j,t−1} | Gross fixed capital formation / GDP. Differenced from t-1 baseline. |
| delta_log_kl | Δlog(K/L) | Cumulated growth | Σ_{s=0}^{h} Δlog(K/L)_{t+s} | Capital per worker growth. K = rnna (PWT constant 2017 prices), L = emp. |
| delta_log_tfp | ΔTFP | Cumulated growth | Σ_{s=0}^{h} Δlog(ctfp)_{t+s} | TFP at current PPPs. Also tested with rtfpna (constant national prices). |
| rgdp_growth | GDP growth | Cumulated growth | Σ_{s=0}^{h} g_{t+s} | Real GDP growth rate, annual. Pre-trends contaminated. |
| mpk_proxy | MPK proxy | Level lead | MPK_{t+h} − MPK_{t−1} | MPK = (1−labsh) × rgdpo / rnna. Differenced from t-1 baseline. |
| delta_log_rnna | Δlog(rnna) | Cumulated growth | Σ_{s=0}^{h} Δlog(rnna)_{t+s} | Capital stock growth. Numerator of K/L decomposition. |
| delta_log_emp | Δlog(emp) | Cumulated growth | Σ_{s=0}^{h} Δlog(emp)_{t+s} | Employment growth. Denominator of K/L decomposition. |
| capital_output_ratio | K/Y ratio | Level lead | (K/Y)_{t+h} − (K/Y)_{t−1} | Capital-output ratio = rnna / rgdpo. |

## Pre-Trend Horizons

We report pre-trends at h = −3, −2, −1 for all outcomes.

**h = −1 omission rationale**: For level variables differenced from $y_{t-1}$,
the h = −1 outcome is mechanically near zero (it measures $y_{t-1} - y_{t-1} = 0$
for the reference period). We therefore focus on h = −3 and h = −2 as informative
pre-trend tests. For cumulated growth variables, h = −1 is the single-period
growth rate at t − 1, which is directly interpretable.

## Horizon Coverage

All outcomes are reported at h = −3, −2, −1, 0, 1, 2, 3, 4, 5.
Full tables with pre-trends are available in Tables 8 and 11.
